Hi, Congrats on your new bass. Hope you're enjoying it.
Personally, I keep a polish cloth in the case with my bass. You can buy them at music stores for a few bucks. After playing, I slip the cloth under the strings and give the fretboard a little shoe shine action, wipe the strings down then the rest of the bass. Keeps it nice and shiny.
When you change strings, you can clean the fretboard with a soft toothbrush and get up close against the frets. I keep my maple fretboard clean and use no cleaners. For my dark fretboards, I keep them clean and polish with a little lemon oil when they start to dry out.
Hope this helps.